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Properties

Label 2-630-105.2-c1-0-9
Degree $2$
Conductor $630$
Sign $0.419 + 0.907i$
Analytic cond. $5.03057$
Root an. cond. $2.24289$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.258 − 0.965i)2-s + (−0.866 + 0.499i)4-s + (1.21 − 1.87i)5-s + (0.576 + 2.58i)7-s + (0.707 + 0.707i)8-s + (−2.12 − 0.686i)10-s + (1.41 − 0.819i)11-s + (1 − i)13-s + (2.34 − 1.22i)14-s + (0.500 − 0.866i)16-s + (2.23 + 0.599i)17-s + (0.274 + 0.158i)19-s + (−0.111 + 2.23i)20-s + (−1.15 − 1.15i)22-s + (8.03 − 2.15i)23-s + ⋯
L(s)  = 1  + (−0.183 − 0.683i)2-s + (−0.433 + 0.249i)4-s + (0.542 − 0.839i)5-s + (0.217 + 0.976i)7-s + (0.249 + 0.249i)8-s + (−0.672 − 0.216i)10-s + (0.427 − 0.246i)11-s + (0.277 − 0.277i)13-s + (0.626 − 0.327i)14-s + (0.125 − 0.216i)16-s + (0.542 + 0.145i)17-s + (0.0629 + 0.0363i)19-s + (−0.0250 + 0.499i)20-s + (−0.246 − 0.246i)22-s + (1.67 − 0.448i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.419 + 0.907i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.419 + 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.419 + 0.907i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ 0.419 + 0.907i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.28695 - 0.823207i\)
\(L(\frac12)\) \(\approx\) \(1.28695 - 0.823207i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 + 0.965i)T \)
3 \( 1 \)
5 \( 1 + (-1.21 + 1.87i)T \)
7 \( 1 + (-0.576 - 2.58i)T \)
good11 \( 1 + (-1.41 + 0.819i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1 + i)T - 13iT^{2} \)
17 \( 1 + (-2.23 - 0.599i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-0.274 - 0.158i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-8.03 + 2.15i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + 1.19T + 29T^{2} \)
31 \( 1 + (-1.5 - 2.59i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (6.83 - 1.83i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + 2.82iT - 41T^{2} \)
43 \( 1 + (-5.63 + 5.63i)T - 43iT^{2} \)
47 \( 1 + (1.63 + 6.10i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-3.01 + 11.2i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-1.04 - 1.80i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.158 + 0.274i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.963 - 3.59i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 7.51iT - 71T^{2} \)
73 \( 1 + (-12.7 - 3.41i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-8.34 - 4.81i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (10.0 + 10.0i)T + 83iT^{2} \)
89 \( 1 + (7.74 - 13.4i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (9.15 + 9.15i)T + 97iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.41087013309677352853783412088, −9.503781335371852670332157521323, −8.747812606928558728206412489934, −8.328717798164755954984876265896, −6.83768395827617197247712519187, −5.57391448090061967930732676900, −5.02095117457315678776053361859, −3.62705814430033687746097127810, −2.36869869100812934103063745324, −1.12748640385813675620546826502, 1.36212800015337185863234031829, 3.13262799847267522665959711624, 4.30162326566505535652095389031, 5.44220308393263243138863421921, 6.49387127083741120599079445515, 7.14678761674291644980261300813, 7.85187096439866552252295132106, 9.156336013817355654427795672379, 9.742958872700809732658912228049, 10.74220607391459228645567819775

Graph of the $Z$-function along the critical line